Non-measurability properties of interpolation vector spaces”, preprint

Abstract. It is known that every dimension group with order-unit of size at most ℵ1 is isomorphic to K0(R) for some locally matricial ring R (in particular, R is von Neumann regular); similarly, every conical refinement monoid with order-unit of size at most ℵ1 is the image of a V-measure in Dobbertin’s sense, the corresponding problems for larger cardinalities being open. We settle these problems here, by showing a general functorial procedure to construct ordered vector spaces with interpolation and order-unit E of cardinality ℵ2 (or whatever larger) with strong non-measurability properties. These properties yield in particular that E+ is not measurable in Dobbertin’s sense, or that E is not isomorphic to the K0 of any von Neumann regular ring, or that the maximal semilattice quotient of E+ is not the range of any weak distributive homomorphism (in E.T. Schmidt’s sense) on any distributive lattice, thus respectively solving problems of Dobbertin, Goodearl and Schmidt. §0. Introduction. Let M be a commutative monoid and let L be a lattice with 0. Say that a map µ: L →M is a measure when µ(0) = 0 and µ(a ∨ b) = µ(a) + µ(b) whenever a ∧ b = 0; furthermore, if L has a largest element 1 and if we have fixed a distinguished element e of M, say that µ is normalized when µ(1) = e. In his paper [5], H. Dobbertin considers Vaught measures